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Grade 7 · Compound probability

Garden measurements: Build a two-stage outcome table

Garden measurements investigation: build a two-stage outcome table. Change the quantities, follow the motion, and explain the result using the displayed mathematical relationship.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Garden measurements: Build a two-stage outcome table. Favorable pairs: 2. Equally likely pairs: 36. P(both): 5.556%Two six-outcome selections: 36 ordered pairs2 / 365.556%Scanning inspects; it does not draw randomly.
Selections are independent and each has six equally likely outcomes. This family models selections with replacement or separate independent devices, not dependent draws without replacement. Scanning is deterministic.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Favorable pairs
2
Equally likely pairs
36
P(both)
5.556%

HD animation studio

From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

Understand what you are seeing

The idea behind the motion.

Two independent six-outcome selections create a 6-by-6 grid of equally likely ordered pairs. If a outcomes are favorable in the first selection and b in the second, the successful rectangle contains ab pairs. Independence is what permits multiplication of the two probabilities. This investigation begins with favorable first outcomes = 1; favorable second outcomes = 2. Treat the named setting as a constructed classroom model, then explain which relationships would still hold if its numbers changed.

A relationship to keep

P(both) = (a/6)(b/6) = ab/36

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Represent the given quantities

    Let columns record the first selection and rows record the second, preserving the order of the pair.

  2. STEP 2

    Follow the changing model

    Playback scans the grid. The amber rectangle identifies pairs where both requirements are met.

  3. STEP 3

    Check and explain the relationship

    Count the favorable pairs and divide by 36; compare that answer with multiplying the two marginal probabilities.

Your turn to explain

Make a prediction. Test your reasoning.

Garden measurements: Independent six-outcome selections have 1 favorable first outcomes and 2 favorable second outcomes. Find P(both).

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

2/36 = 5.556%; multiply the two independent probabilities.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 7 · Statistics

List compound outcomes before counting

A systematic outcome table prevents missing or duplicating possibilities in a two-stage experiment.